Theorem 3Let g be continuous at a and let f be continuous at g(a), then f◦g iscontinuous at a.Proof: Use rule for composition of limits — for you to do!Combining Continuous Functions5Example 4Letf:R→Rbe given byf(x) =braceleftBiggcos(ax)forx≤πbxforx> π.For what values ofaandbwillfbe continuous?From the definition offwe see thatfis continuous whenxnegationslash=π. In fact,sincef(x) =cos(ax)whenx< π, and the cosine function iscontinuous,fis continuous.

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